Complex logarithm

One may define the natural logarithm also for all non-zero complex numbers z, but it is usually denoted log(z) for mostly two reasons: To distinguish the function from the usual ln, and because no base other than the natural base is used in the complex domain, making it the only Log function for complex numbers. The first Taylor expansion

\ln(1+z)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} z^n = z - \frac{z^2}{2} + \frac{z^3}{3} - \cdots \quad{\rm for}\quad \left|z\right|<1,

remains valid for all complex numbers z with absolute value less than 1. If the non-zero complex number z is expressed in polar coordinates as

z = r e^{i \phi}

with r > 0 and

-\pi < \phi < \pi

, then

\log(z) = \ln(r) + i\phi \,

where ln(r) is the usual natural logarithm of a real number.

So defined, log is holomorphic for all complex numbers which are not non-positive reals, and it has the property

e^{\log(z)} = z \,

for all nonzero z. One has to be careful, because several properties familiar from the real logarithm are no longer valid for this complex extension. For example, log(e^z) does not always equal z, and log(zw) does not always equal log(z) + log(w).

A somewhat more natural definition of log(z) interprets it as a multi-valued function: for $z = r e^{i \phi}$ we set

\log(z) = \ln(r) + i(\phi + 2 \pi k) \,

where k is any integer. This is the set of all complex numbers u for which $e^u = z$ , because $e^{2\pi i} = 1$ (see Euler's identity).

The preferred way to deal with multivalued functions like this in complex analysis is via Riemann surfaces: the function log is then not defined on the complex plane but instead on a suitable Riemann surface having countably many "leaves" and the values of the function differ by 2πi from leaf to leaf.

Gourt Home::Gourt :: Complex logarithm

See also